The first Pixar logo was custom designed by John Lasseter, working with a stone cutter (I'll try to find his name) to design the font, and used on the face of what eventually became known as the Pixar Image Computer.

We nicknamed this version "the BSD" (Beveled Square with a Dent), a play on the Berkeley Software Distribution version of Unix we used.

It has since been simplified by removing the diamonds between the letters and, now that I look at it, tweaking the font:

The name is a made-up noun, akin to Kodak or Xerox, and is meant to evoke pixels while sounding vaguely like a Spanish verb. It was invented to name the machine, and was later appropriated to become the name of the company when we spun off from Lucasfilm.

The capital letters have confused many into thinking it's PIXAR, but it's properly Pixar. Only the logo is all caps.

Now, if by "Pixar Logo" you mean the studio logo animation featuring the hopping lamp, that was invented at the last minute by Pete Docter right before the release of Toy Story. The letters from the logo were modeled in 3D, and Pete used the lamp character from our 1986 short film, Luxo Jr.

Since Pete did it on his personal machine and his home directory, it was never properly entered as a studio asset, and the original animation was eventually lost. What you see on Pixar films now is a recreation of Pete's animation done by, if memory serves, Ginny Santos.